What is the slope of tangent at (2, 4) to the curve y = x²?

Explanation

To find the slope of the tangent to the curve $y=x^2$ at the point $(2,4)$, we compute the derivative of $y$ with respect to $x$, which gives the slope of the tangent at any point on the curve.

1. Differentiate $y=x^2$:

   $$\frac{\mathrm{<b><font\; color="\#0000ff">d</font></b>}\mathrm{<b><font\; color="\#0000ff">y</font></b>}}{\mathrm{<b><font\; color="\#0000ff">d</font></b>}\mathrm{<b><font\; color="\#0000ff">x</font></b>}}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}\mathrm{<b><font\; color="\#0000ff">x</font></b>}$$

2. Evaluate the derivative at $x=2$:

   $${\frac{\mathrm{<b><font\; color="\#0000ff">d</font></b>}\mathrm{<b><font\; color="\#0000ff">y</font></b>}}{\mathrm{<b><font\; color="\#0000ff">d</font></b>}\mathrm{<b><font\; color="\#0000ff">x</font></b>}}<b><font\; color="\#0000ff">\mid </font></b>}_{\mathrm{<b><font\; color="\#0000ff">x</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}<b><font\; color="\#0000ff">\times </font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">4</font></b>}$$

Therefore, the slope of the tangent at $(2,4)$ is $4$.

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